Exact calculation of the probabilities of rare events in cluster-cluster aggregation

TL;DR

This paper introduces an action formalism to precisely compute the probabilities of rare events in cluster-cluster aggregation, providing exact rate functions and trajectories for various kernels, validated by simulations.

Contribution

It develops a general action formalism and establishes a large deviation principle for arbitrary collision kernels, with exact calculations for specific kernels and analysis of rate function properties.

Findings

Exact rate functions for constant, sum, and product kernels.

Discontinuity in the second derivative of the rate function for the product kernel.

Theoretical results are confirmed by tailored simulations.

Abstract

We develop an action formalism to calculate probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels and establish a pathwise large deviation principle with total mass being the rate. As an application, the rate function for the number of surviving particles as well as the optimal evolution trajectory are calculated exactly for the constant, sum and product kernels. For the product kernel, we argue that the second derivative of the rate function has a discontinuity. The theoretical results agree with simulations tailored to the calculation of rare events.